# BHARDWAJ CLASSES

## CBSE CLASS 12 MATHS SAMPLE PAPER 2018/ PRACTICE PAPER 2018

Practice Paper – 1
Time Allowed : 3 Hours                                                                                                Maximum Marks : 100
General Instructions :
(i) All questions are compulsory.
(ii) This model test paper contains 29 questions.
(iii) Questions 1 to 4 in section – A are very short answers type questions carrying mark each.
(iv)Questions 5 to 12 in section – B are short answer type questions carrying 2 marks each.
(v) Questions 13 to 23 in section – C are long answer – I type questions carrying 4 marks each.
(vi) Questions 24 to 29 in section – D are long answer – II type questions carrying 6 marks each.
Section – A
Questions 1 to 4 Carry 1 mark each.

1. Show that the relation R in the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is reflexive.
2. If $f:R\to R\text{\hspace{0.17em}}and\text{\hspace{0.17em}}f\left(x\right)=\frac{7x+3}{5}$is an invertible function, Find ${f}^{-1}\left(x\right)$
3. If $\stackrel{\to }{a}=2\stackrel{^}{i}+\stackrel{^}{j}+5\stackrel{^}{k}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\stackrel{\to }{b}=\stackrel{^}{i}-\stackrel{^}{j}+4\stackrel{^}{k}$, find $\stackrel{\to }{a}×\stackrel{\to }{b}$
4. If matrix A = $\left[\begin{array}{c}3\\ 1\\ 2\end{array}\right]$ and matrix B = [2 0 5], find AB.
Section – B
Questions 5 to 12 carry 2 marks each.

5. Show that $\mathrm{cos}\left[\frac{\pi }{2}-{\mathrm{tan}}^{-1}x\right]=\frac{x}{\sqrt{1-{x}^{2}}}$
6. If y is the angle between two unit vectors $\stackrel{^}{a}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\stackrel{^}{b}$ , then show that :$\mathrm{sin}\left(\frac{y}{2}\right)=\frac{1}{2}|\stackrel{^}{a}-\stackrel{^}{b}|$
7. Construct a matrix A = ${\left[{a}_{ij}\right]}_{3×3}={\left(3\right)}^{i}+{\left(2\right)}^{j}$
8. Evaluate : $\int {\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)}^{2}dx$
9. If ${x}^{2}+{y}^{2}+xy+5=0,\text{\hspace{0.17em}}find\text{\hspace{0.17em}}\frac{dy}{dx}$
10. Solve the differential equation : $\frac{dy}{dx}=1+x+y+xy$
11. If P(A) = 9/20 , P(B) = 8/15 , and P(A U B) = 47/60 , Find P(A/B)
12. Find the equation of a tangent to the curve: $y=3{x}^{2}+5\text{\hspace{0.17em}}$ at a point (1, 8)
Section - C
Questions 13 to 23 carry 4 marks each.
13. Find the value of k so that the function given below becomes continuous at x = -1.
14. If $A=\left[\begin{array}{cc}3& 8\\ 2& 1\end{array}\right]\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{A}^{2}-mA+nI=0$ , find the value of m and n and hence find ${A}^{-1}$
15. If $y={\mathrm{tan}}^{-1}\left[\frac{\sqrt{1+\mathrm{cos}x}-\sqrt{1-\mathrm{cos}x}}{\sqrt{1+cos\text{\hspace{0.17em}}x}+\sqrt{1-\mathrm{cos}x}}\right],find\frac{dy}{dx}$
16. The length x cm of a rectangle decreases at the rate of 5 cm/min and width y cm increases at the rate of 4 cm/min. Find the rate of change of perimeter and the area of the rectangle when x = 8 cm and y = 6 cm.
OR
Verify Rolle’s theorem for the function$f\left(x\right)={x}^{3}-7{x}^{2}+16x-12\text{\hspace{0.17em}}on\left[2,3\right]$
17. A manufacturer can sell x items at a prize of Rs $\left(5-\frac{x}{100}\right)$each. The cost price of x items is Rs$\left(\frac{x}{5}+500\right)$ . Find the number of items , he should sell to earn maximum profits.
18. Find $\int \frac{\left(\mathrm{cos}x\right)dx}{\left({\mathrm{sin}}^{2}x+1\right)\left({\mathrm{sin}}^{2}x+2\right)}$
19. Show that the differential equation of all the parabolas
${y}^{2}=4a\left(x-b\right)\text{\hspace{0.17em}}is\text{\hspace{0.17em}}given\text{\hspace{0.17em}}by\text{\hspace{0.17em}}\text{\hspace{0.17em}}y\left(\frac{{d}^{2}y}{d{x}^{2}}\right)+{\left(\frac{dy}{dx}\right)}^{2}=0$
OR
Solve the differential equation ${\mathrm{sin}}^{-1}\left(\frac{dy}{dx}\right)=\left(x+y\right)$
20. For any vector $\stackrel{\to }{r}$show that $\stackrel{\to }{r}=\left(\stackrel{\to }{r}.\stackrel{^}{i}\right)\stackrel{^}{i}+\left(\stackrel{\to }{r}.\stackrel{^}{j}\right)\stackrel{^}{j}+\left(\stackrel{\to }{r}.\stackrel{^}{k}\right)\stackrel{^}{k}$
21. You are given a point A( 1, 6, 3) and a line BC:$\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ . Find the distance of the point A from the line BC .
22. One card from a pack of 52 cards is lost. From the remaining pack of cards one card is drawn and is found to be diamond. Find the probability that the lost card is also diamond.
23. A box contains 4 white and 3 black balls. Two balls are drawn with the replacements. Find the probability distribution, mean and variance of white balls.
Section - D
Questions 24 to 29 carry 6 marks each.
24. R is a relation defined on real numbers as (a, b) R(c, d) if ad(b + c) = bc(a + d). Show that R is an equivalence relation.
OR
Show that ${\mathrm{tan}}^{-1}\left[\frac{\sqrt{1+{x}^{2}}+\sqrt{1-{x}^{2}}}{\sqrt{1+{x}^{2}}-\sqrt{1-{x}^{2}}}\right]=\frac{\pi }{4}+\frac{1}{2}{\mathrm{cos}}^{-1}\left({x}^{2}\right)$
25. Using properties of determinants, show that:$\Delta =|\begin{array}{ccc}a-b-c& 2a& 2a\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}|=2{\left(a+b+c\right)}^{3}$
OR
$if\text{\hspace{0.17em}}A=\left[\begin{array}{ccc}5& 0& 4\\ 2& 3& 2\\ 1& 2& 1\end{array}\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\text{\hspace{0.17em}}B=\left[\begin{array}{ccc}1& 3& 3\\ 1& 4& 3\\ 1& 3& 4\end{array}\right]\text{\hspace{0.17em}}find\text{\hspace{0.17em}}{\left(AB\right)}^{-1}$
26. Draw the rough graph of the curves: $y={x}^{2}-2x-3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\text{\hspace{0.17em}}x+y=9$ and using integration, find the area between them.
27. Using limit of sum methods, evaluate $\underset{2}{\overset{5}{\int }}\left(3{x}^{2}+2x+7\right)\text{\hspace{0.17em}}dx$
OR
Evaluate$\underset{0}{\overset{\frac{\pi }{4}}{\int }}\frac{\mathrm{sin}x\mathrm{cos}x\text{\hspace{0.17em}}\text{\hspace{0.17em}}dx}{{\mathrm{sin}}^{4}x+{\mathrm{cos}}^{4}x}$
28. Find the length and foot of the perpendicular drawn from a point P(7, 14, 5) to the plane 2x + 4y –z = 2. Also find the image of the point P in the plane.
29. A producer has 30 and 17 units of labour and capital respectively which he can use to produce two types of goods X and Y. To produce one unit of X, 2 units of labour and 3 units of capital are required. Similarly, 3 units of labour and 1 unit of capital is required to produce one unit of Y. If X and Y are priced at Rs. 100 and Rs. 120 per unit respectively, how should the producer use his resources to maximize the total revenue? Solve the problem graphically.

Also Read : CBSE Class 12 Maths Sample Paper 2018